Alma Mater Studiorum · Università di Bologna
SCUOLA DI SCIENZE
Corso di Laurea Magistrale in Matematica
GEOMETRIC PROPERTIES OF
2-DIMENSIONAL MINIMAL
SURFACES
IN A SUB-RIEMANNIAN
MANIFOLD WHICH MODELS
THE VISUAL CORTEX
Tesi di Laurea in Analisi Geometrica
Relatore:
Chiar.mo Prof.
Giovanna Citti
Correlatore:
Chiar.mo Prof.
Manuel Ritoré
Presentata da:
Gianmarco Giovannardi
II Sessione
Anno Accademico 2015-2016
Abstract
In this paper we study the notion of degree for submanifolds embedded in an equiregular sub-Riemannian manifold and we provide the definition of their as-sociated area functional. In this setting we prove that the Hausdorff dimension of a submanifold coincides with its degree, as stated by Gromov in [19]. Using these general definitions we compute the first variation for surfaces embedded in low dimensional manifolds and we obtain the partial differential equation associ-ated to minimal surfaces. These minimal surfaces have several applications in the neurogeometry of the visual cortex.
Sommario
In questa tesi studiamo la nozione di grado per sottovarietà immerse in una varietà sub-riemanniana e forniamo la definizione del funzionale dell’area ad esse associato. In questo ambiente proviamo che la dimensione di Hausdorff di una sottovarietà coincide effettivamente con il suo grado, come affermò Gromov nel suo lavoro [19]. Utilizzando queste definizioni generali calcoliamo la variazione prima dell’area per sottovarietà immerse in varietà di dimensione bassa e otteniamo l’equazione alle derivate parziali associata alle superfici minime. Queste superfici minime hanno diverse applicazioni nella neurogeometria della corteccia celebrale.
Introduction
In the last years variational problems in sub-Riemannian geometry have become the object of many studies. In particular, Pauls in [30], Hurtado, Ritoré and Rosales in [23], Capogna, Citti and Manfredini in [4] , Cheng Hwang and Yang in [6] deal with minimal surfaces in Heisenberg group H1. Moreover, in [1,5,15–17] these problems in a more general setting, as contact sub-Riemannian manifolds or Caront groups, are studied
Our aim in this master thesis is to study the area functional for a smooth sub-manifold embedded in a sub-Riemannian sub-manifold. First of all, we recall that a sub-Riemannian manifold N is a smooth manifold endowed with a distribution H which is a subbundle of the tangent bundle and a horizontal metric h defined only on the distribution. In the present work a crucial assumption is that distribution H verifies the celebrated Hörmander rank condition at each point p in N. Let X1, · · · , Xk be a local frame forH where k = dim(H), we say that H verifies the Hörmander rank condition if the vector fields X1, · · · , Xk and all their commuta-tors of any order generate all the tangent space. This condition has been deeply studied after the first studies by Hörmander in [21], Rothschild and Stein in [34], Nagel, Stein and Wainger in [29] and Montgomery in [28]. Under this condition, Chow’s Theorem implies that any couple of points can be connected by horizontal curves (see [7]). Thus, it is possible to define the Carnot-Carathéodory distance on N as the infimum of length of horizontal curves joining two given points. Iterated Lie brackets of horizontal vector fields generate a flag of subbundles
(1) H ⊂ H2 ⊂ · · · ⊂Hr ⊂ · · · ⊂Hs = T N, where
Hr+1 =Hr+ [H, Hr], [H, Hk] = {[X, Y ] : X ∈H, Y ∈ Hk}.
Moreover, the integer list (n1(p), · · · , nr(p)) where ni(p) = dim(Hi) is called the growth vector of H at p.
Our goal is to give a suitable definition of the area for a submanifold em-bedded in N . Franchi, Serapioni and Serra Cassano deeply studied regular sur-faces in sub-Riemannian structures (see [12–14]). In the present work, we follow Magnani-Vittone [27] and Le Donne-Magnani [24] approach, consisting in con-sidering submanifold, regular in Euclidean sense, and its associated degree. They show that the area of a submanifold in the Engel’s group in [24] and in stratified
Carnot groups in [27] and [26] is strictly connected to this notion of degree of a submanifold. Here, we consider only equiregular sub-Riemannian manifolds where the growth vector is pointwise constant in N . We define the degree on an adapted basis (X1, · · · , Xn) to flag (1) that exists by the Hörmander rank condition. We say that Xi has degree r if Xi lies in Hr but Xi does not belong to Hr−1. Then, taking a m-vector XJ = Xj1∧ · · · ∧ Xjm we set the degree d(X
J) of XJ as the sum of degrees of each vector in the wedge product
d(XJ) = d(Xj1) + · · · + d(Xjm).
Essentially, the degree is a pointwise property of the tangent space at p , indeed the m-vector τΣ(p) which represents the tangent space of the submanifold is a linear combination of m-vectors as XJ
τΣ(p) = X
J
τJXJ|p,
thanks to the linearity of the wedge product. Automatically, the degree of τΣ(p) is the maximum integer d(XJ) such that τ
J is different from zero. This definition of degree is equivalent to Gromov’s definition of degree (see [19]). Basically the degree measures the intersection between each layer of the flag and the tangent space therefore it is strictly connected to the geometrical structure submanifold inherited by the ambient sub-Riemannian structure.
When the ambient space is a Riemannian manifold equipped with a metric g it is clear how we define the area of a submanifold, area(Σ, g), using the Riemannian area element depending on the metric g. When we consider a sub-Riemannian manifold there is a lack of a metric on the tangent bundle, since there exists only a horizontal metric h on the subbundle given by the distribution. In order to give the definition of area we extend the horizontal metric h to a Riemannian metric g such that g makes Hi = Hi+1/Hi spaces orthogonal and g|H = h. Then it is natural to construct a sequence of metrics gr defined on the basis (X1, · · · , Xn) as (2) gr(Xi, Xj) =
rd(Xi)+d(Xj )−22
−1
g(Xi, Xj) i, j = 1, · · · , n.
Clearly, the restriction of gr to the distributionH is equal to the horizontal metric h, gr|H = h and when we let r tend to zero the metric blows up out of H. Thus, the metric gr in the limit provides a good representation of h and shows that only horizontal curves are allowed. Indeed, a curve not tangent to the distribution at each point has infinite length
Since we have a sequence that converges to the sub-Riemannian metric, we define the sub-Riemannian area for a m-dimensional submanifold Σ of degree d embedded in a sub-Riemannian structure by
(3) A(Σ) = lim
r→0 r
d−m
INTRODUCTION 9
Moreover, Gromov in his book [19] claimed that the Hausdorff dimension of a submanifold M embedded in an equiregular sub-Riemannian manifold N is its degree but he did not prove it. Ghezzi and Jean in [18] provided a proof of this assertion only for a a strongly equiregular submanifold (see definition 2.10).
In the present work we provide a different proof for Ghezzi and Jean’s result and then we give a proof Gromov’s affirmation. Basically, the intersection between the tangent space of M and each layer of the stratification generates a flag and the strongly equiregular assumption assures that the dimension of each space of the flag is constant pointwise in M . We consider privileged coordinates adapted to the flag given by the exponential map in a neighborhood of a point p. Thanks to the Ball-Box Theorem, balls are equivalent to boxes (see [28, Theorem 2.4.2]). Thus we cover the intersection between boxes and submanifold M , which in privileged coordinates is
Boxw(r0) ∩ {x ∈ Rn : xm+1 = · · · = xn= 0}
with boxes of size 1/k, thus we obtain that the Hausdorff dimension is equal to d. Then we realize that the degree of vector fields is lower semicontinuos, therefore, if we fix the degree of a submanifold M , a simple vector fields of m-vector tangent to M can not switch its degree in a neighborhood of a point p. Hence, the flag used in the previous proof has locally constant dimension, then we can apply the precedent argument to a neighborhood of a point p.
In the Euclidean space a standard definition of the mean curvature for a sub-manifold is obtained by the first variation of the area functional. Nowadays, a central problem in Geometric Analysis is to provide a good definition of the mean curvature in different settings, as in sub-Riemannian geometry, by computing the first variation of the area functional. Our principal motivation to minimize the area functional came from the neurogeometry of the brain where we learnt that the order of the mean curvature operator could be greater that two.
A mathematical model of simple cells S of the visual cortex V1 using the sub-Riemannian geometry of the roto-translational Lie group was proposed by Citti and Sarti (see [8], [9]). In their work, the perceptual completion is obtained through minimal surfaces and therefore they studied the regularity and foliation properties of minimal surfaces in S= E(2). Their techniques have several applica-tions in image completion. In [11] it was conjectured that endstopping cells E are sensible to curvature and a sub-Riemannian structure modelling their structure was proposed in [31]. In this work we shall consider an extension of the results of [9] to this family of cells. The space S will be identified with R2× S1. We shall consider the distribution generated by the vector fields
and we shall compute the first variation of the area for a θ-graph in S to obtain the minimal surface equation
X X(θ)
p1 + X(θ)2 !
= 0.
This equation along characteristic curves is equivalent to θ00= 0, where the deriva-tive0 is taken with respect to the arc-length parameter of the curve, see [15], [16]. Adding an additional variable, the curvature, to the three-dimensional space S we obtain the space E= R2× S1× R, where we consider the distribution generated by
X1 = cos(θ)∂x+ sin(θ)∂y+ k∂θ and X2 = ∂k,
(see also [31]). In this setting we are interested in (θ, κ)-graphs which are 2-dimensional surfaces. Notice that a (θ, k)-graph has a foliation property if and only if the equation X1(θ) = κ holds. Moreover, this condition implies that the degree of the surface is four. Thus applying definition (3) we obtain
A(Σ) = Z
Ω p
1 + X1(κ)2 dx dy.
Critical points of this area functional satisfy the following minimal PDE equation (4) X4(X1(θ)) + X1(X4(θ))) + X1(g)X4(θ) + X12(g) = 0,
where we set
g = X1(κ) p1 + X1(κ)2
.
Notice that equation (4) is a third-order partial differential equation which we attempt to read along characteristic curves as we do in S. However, there is X4(θ) term which corresponds to the derivative in the direction perpendicular to the tangent direction of characteristic curves projected on the retinal plane. Therefore, we consider different horizontal metrics h1, h2, h3 that imply different minimal PDEs, but we have not succeeded in reading these equations along characteristic curves.
In Chapter1we provide the definition of a sub-Riemannian manifold, of a dis-tribution, with its natural Carnot-Carathéodory distance, and some basic notions about the geodesic equation. Then we compare the involutive condition that im-plies the Frobenius Theorem with the Hörmander rank condition that imim-plies the Chow Theorem. Furthermore, we define the exponential map, Lie derivative, reg-ular surfaces and we report the Rothschild-Stein’s theorem that assures that the tangent space to a sub-Riemannian structure is a Carnot group. In the last section we supply some example of sub-Riemannian manifold as the rototraslation group S and 2-jet space E, then we show that tangent space to S and E are respectively the Heisenberg group and the Engel’s group.
INTRODUCTION 11
Chapter 2 focuses on the degree and the definition of area functional for an embedded submanifold in an equiregular sub-Riemannian manifold. In this chapter we find condition under which the area functional is independent of the extension of the horizontal metric h up to a positive constant. Moreover, we show that this general definition of area corresponds to the area of hypersurfaces in the Heisenberg group. Then we study the geometry of surfaces and the area functional in S and E. In conclusion we prove the Gromov’s conjecture about the Hausdorff dimension of a submanifold.
Finally, in Chapter3we compute the first variation of the area functional for a surface in S and E and we obtain the PDEs associated to the minimal surfaces. In E we notice that only variations preserving degree four are allowed, otherwise the area functional changes. Then we study general variations preserving the degree d of a submanifold in a general equiregular sub-Riemannian manifold and we obtain a PDE system of equations where the coefficients of the vector field X inducing the variation are involved. This system restricts the range of permitted variations.
Contents
Abstract 3
Sommario 5
Introduction 7
Chapter 1. Introduction to sub-Riemannian geometry 15
1. Definition of sub-Riemannian structure 15
2. Differential Operators 22
3. Examples and applications 25
Chapter 2. Area in a sub-Riemannian manifold 35
1. Degree of a submanifold in a sub-Riemannian manifold 37
2. Equivalence between our degree and Gromov’s degree 38
3. Semicontinuity of the degree 40
4. Sub-Riemaniann area of a submanifold 41
5. An interesting case of sub-Riemanian area 43
6. Area of a hypersurface in the Heisenberg group 45
7. Area of a surface in a sub-Riemannian manifold 48
8. Sub-Riemannian area of a surface in S 49
9. Geometry of surfaces in 4-dimensional sub-Riemannian manifolds 50
10. Hausdorff dimension 58
Chapter 3. First variation of area functional 63
1. Minimal surface equation in S 64
2. First variation of the area with fixed degree in E 66
3. First variation of the area in a different manifold 72
4. First variation of the area of a submanifold 76
5. PDE system restrictions for a graph 80
Bibliography 89
CHAPTER 1
Introduction to sub-Riemannian geometry
1. Definition of sub-Riemannian structure Following Montgomery, we give the following definition.
Definition 1.1. A sub-Riemannian geometry on a smooth manifold N consists of a distribution H ⊂ T N, which is a vector subbundle of the tangent bundle of N , together with a fiber inner-product h on this subbundle.
We will call H the horizontal distribution and the inner product h will be referred to as the horizontal metric. A vector field is horizontal if it is everywhere tangent to H. A C1 curve in N is said to be horizontal if the tangent vector is horizontal at every point. Let γ : [a, b] → N be a smooth horizontal curve, we define the length of γ by
(5) l(γ) =
Z b
a p
h( ˙γ(t), ˙γ(t)) dt.
Notice that we define the length only for horizontal curves, where the inner product exists. We use the length to define the distance between two points p and q in N , as in Riemannian geometry:
(6) d(p, q) = inf{l(γ) : γ is horizontal curve such that γ(a) = p, γ(b) = q}. If there is not a horizontal curve which joints p and q, we set that the distance is infinite. This is the well-known Carnot-Carathéodory distance, for brevity C-C distance.
Now, it is natural to whether there is a condition that assures that the distance between each points p and q in N is always finite. In other words, given every pair of points p and q in N , we want to know under which condition there exists a horizontal curve γ such that γ(a) = p and γ(b) = q. In order to answer this question we have to introduce the Hörmander rank condition. Given a distribution H of dimension k with inner product h, we can consider an orthonormal local frame X1, · · · , Xk. On the other hand, we can give X1, · · · , Xk and say that H is the distribution generated by these vector fields and set a horizontal inner product such that X1, · · · , Xk is an orthonormal basis. We prefer the first approach, because in visual cortex cases we know the vector fields which generate the distribution but we have doubts about the choice of horizontal metric h.
Below, we recall some well-known definitions and theorems.
Definition 1.2. Let ϕ : M → N be smooth function. The vector fields X on M and Y on N are called ϕ-related if dϕ ◦ X = Y ◦ ϕ.
Definition 1.3. Let X, Y be first-order regular differential operators (i.e. vec-tor fields). Their commutavec-tor is defined by
[X, Y ] = XY − Y X
and it is also a first-order differential operator. We define the Lie algebra generated by X1, · · · , Xk and denote it by
L(X1, · · · , Xk)
the linear span of the operators X1, · · · , Xk and their commutators of any order. We set that a commutator has degree r,
d(X) = r if X = [· · · [Xi1, Xi2], · · · , Xir] = Ad(Xi1, · · · , Xir)
with i1, · · · , ir ∈ {1, · · · , k}.
In order to understand how the Hörmander rank condition is connected to the connectivity it is useful to remind what is an involutive distribution.
Definition 1.4. A smooth distribution H is called involutive if [X, Y ] ∈ H whenever X and Y are smooth vector fields lying in H. In other words, let X1, · · · , Xk be a local frame of H, then
L(X1, · · · , Xk) = span{X1, · · · , Xk}.
Definition 1.5. Let (M, ϕ) be a submanifold of N . We say that M is an integral manifold of a distribution H on N if
dϕ(TpM ) = H|ϕ(p) for each p ∈ M.
Theorem 1.1 (Frobenius). Let D be a k-dimensional smooth distribution on N . Then, D is involutive if and only if there exists an integral manifold of D passing through every point of N .
Proof. Here, we prove only that the existence of an integral manifold of D implies that D is involutive. We have to prove that [X, Y ] ∈ D whenever X and Y are smooth vector fields lying in D. By hypothesis, let (M, ϕ) be an integral manifold ofD through p = ϕ(m), therefore
dϕ : TmM →D|ϕ(m)
is a isomorphism. Then, there exist vector fields ¯X and ¯Y such that dϕ( ¯X|m) = X|ϕ(m), dϕ( ¯Y|m) = Y|ϕ(m).
Moreover, ¯X and ¯Y are smooth and ϕ-related. By [36, Proposition 1.55], which assures that if ¯X and ¯Y are ϕ-related then [ ¯X, ¯Y ] and [X, Y ] are ϕ-related, we have [X, Y ] = dϕ([ ¯X, ¯Y ]) ∈ D. The proof of other implication is done by induction on the dimension of the distribution, for further details see [36, Theorem 1.60].
1. DEFINITION OF SUB-RIEMANNIAN STRUCTURE 17
When a k-dimensional distribution H on N is involutive and we consider a point p, by Frobenius Theorem, we have that there exists an integral manifold passing through p. This integral manifold of dimension k, called the leaf through p, is generated by the set of horizontal paths through the fixed point p. Therefore, if q does not lie in the leaf of p we can not connect p and q by a horizontal curve. Thus, the distance between p and q would be infinite. Opposite to involutive distributions we have the ones that verify Hörmander rank condition, also known as bracket-generating distributions.
Definition 1.6. We say that a distribution H on a n-dimensional manifold N verifies the Hörmander rank condition if any local frame {X1, · · · , Xk} for H satisfies
dim(L(X1, · · · , Xk))(p) = n ∀ p ∈ N.
In other words, the Lie algebra generated by X1, · · · , Xk is all the tangent bundle. Let s be the smallest natural number such that X1, · · · , Xkand their commutators of degree smaller that or equal to s span the all tangent space. We will call s the step at a point p and the local basis X1, · · · , Xk, Xk+1, · · · , Xn made out of com-mutators of X1, · · · , Xkis chosen such that, for every point, the local homogeneous dimension (7) Q = n X j=1 d(Xj) is minimal.
Theorem 1.2 (Chow). If a distribution H ⊂ T N verifies Hörmander rank condition, then the set of points that can be connected to p in N by a horizontal path is the connected component of N containing p.
We suggest the reader to see [28, 2.2] for the heuristic Hermman’s proof of Chow Theorem or [28, 2.4] for a standard proof.
Example 1.1. Let us show two examples of sub-Riemannian manifolds that do not verify the Hörmander rank condition at each point. Let R2 be the plane and let
f (x) = (
0 if x = 0 e−x21 if x 6= 0
be a C∞ function such that f(n)(0) = 0 for all n = 1, 2, 3, · · · . We consider the distribution generated by the vector fields
X1 = 1 0 , X2 = 0 f (x) .
If we consider a point p = (x, y) such that x 6= 0, X1 and X2 generate all the tangent space. Let p = (x, y) be a point in R2 such that x = 0, we have X
Moreover, all brackets of length n are equal to zero at p = (0, y), the proof follows from induction. At first for n = 1, we have
[X1, X2](0, y) = lim h→0 e−h21 − 0 h ! ∂y = lim x→∞x e −x2 ∂y = 0. Now, we assume that
Xn+2 := [X1, · · · , [X1, X2], · · · ] = f(n)(0)∂y = 0. Moreover, since it holds
f(n)(x) = P2n−2(x) e −1 x2 x3n , we have Xn+3(p) = [X1, Xn+2](p) = lim h→0 P2n−2(h) e− 1 h2 h3n − 0 h ∂y|p = lim x→∞Pk(x)e −x2 ∂y|p = 0.
Hence, this distribution does not verify the Hörmander rank condition. However, it is possible to possible to connect all possible points with integral curves. Indeed, the only problem could be at point (0, y), but it is possible to leave this point by the vector field X1 and there we can reach every point.
Thus, Hörmander rank condition assures that the Carnot-Carathéodory dis-tance is finite. Furthermore, in definition (6) there is an infimum over all possible horizontal path with fixed endponits. A natural question may be when this infi-mum is reached. To answer this question we have to minimize the length functional l(γ) defined in (5) over all possible horizontal curves γ with fixed endponits. As in Riemannian geometry minimizing the length functional is equal to minimizing the energy functional
(8) E(γ) = Z γ 1 2k ˙γk 2 , where k ˙γk2 = h( ˙γ, ˙γ). Indeed, thanks to Cauchy-Schwarz inequality, we have
l(γ) = Z γ k ˙γk · 1 6 s Z γ k ˙γk2 √b − a = p2 E(γ) √b − a
with equality if and only if k ˙γk = c. We denote ¯α the curve α covered at constant speed k ˙¯αk = c. Therefore, if γ minimizes E and η is another horizontal path connecting p and q, it follows
1. DEFINITION OF SUB-RIEMANNIAN STRUCTURE 19
then γ minimizes the length functional l. On the other hand, if γ minimizes l and η is another horizontal path connecting p and q, we have
p
2 E(¯γ) √b − a = l(¯γ) = l(γ) 6 l(η) 6p2 E(η)√b − a, then γ minimizes the energy functional E.
Definition 1.7. An absolutely continuous horizontal path that realizes the distance between two points is called a minimizing geodesic or simply a geodesic. In a sub-Riemannian setting there is a lack of a covariant two-tensor like the Riemannian metric in Riemannian geometry. However, it is possible to define a contravariant symmetric two-tensor, a section of T N ⊗ T N . This tensor is called the cometric and has rank k, the dimension of the distribution. In [28, 1.5] Montgomery shows that from this cometric acting on covectors it is possible to define the sub-Riemannian Hamiltonian or kinetic energy
H : T∗N → R, H(q, p) = 1
2((p, p))q,
where q ∈ N , p ∈ Tq∗N and ((·, ·)) is the cometric, such that 12k ˙γk2 = H(q, p). Here, q = γ(t) and p such that ˙γ(t) = βγ(t)(p). Minimizing the energy functional E we obtain the Hamiltonian differential equations
(9) ˙xi = ∂H
∂pi
, p˙i = − ∂H ∂xi
These Hamiltonian differential equations are called the normal geodetic equations. The following theorem holds
Theorem 1.3. Let (γ(t), p(t)) be a solution to Hamilton’s differential equations on T∗N for the sub-Riemannian Hamiltonian H and let γ(t) be its projection to N . Then every sufficiently short arc of γ is a minimizing sub-Riemannian geodesic. Moreover γ is the unique minimizing geodesic joining its endpoints.
Definition 1.8. The projection γ of the previous theorem is called a normal geodesics.
Remark 1.1. There are sub-Riemannian manifolds that admit minimizing geodesics that do not solve the Hamilton’s differential equations. These geodesics are called singular geodesics. Montgomery in his book [28] deeply studied this topic.
It is also worth mentioning another important theorem which shows that the topology induced by the Carnot-Carathéodory distance has the same topology of the smooth manifold N .
Theorem 1.4. If the distribution H on N satisfies the Hörmander rank con-dition, then the topology on N induced by the Carnot-Carathéodory distance is the usual manifold topology.
The proof follow from the well-known Ball-Box Theorem 2.2 (see [28, 2.4 and 2.5]).
Remark 1.2. In the present work we follow Montgomery’s approach. Some authors include the Hörmander rank condition in the definition of sub-Riemannian manifold.
1.1. Lie group and Carnot group. Let (G, ·) be a Lie group (see [36, Definition 3.1]) and let g be its Lie algebra. We consider V ⊂ g a linear subspace of the Lie algebra. We can see the Lie Algebra as the space of all left invariant vector fields, i.e. lg-related to themselves where the left translation by g in G is
lg(h) = g · h.
In this way, V is a left invariant distribution and the Hörmander rank condition corresponds to the fact that V Lie-generates g. If we set an inner product h on V we obtain a sub-Riemannian metric. Therefore (G, V, h) has a sub-Riemannian structure.
Definition 1.9. We say that G is a graded nilpotent Lie group if the Lie algebra g has the form
g= V1⊕ V2⊕ · · · ⊕ Vs
where [Vi, Vj] = Vi+j and Vr= 0 if r > s. Therefore, all iterated brackets of length r > s are zero. If we define an inner product h on V1 and we suppose that V1 Lie-generates g, then we obtain a sub-Riemannian manifold (G, V1, h). We will call this structure a Carnot group.
1.2. Exponential mapping. Another tool we need in this work is the expo-nential mapping induced by vector fields. As we need local results, we work in an open set (U, ψ) of N . Therefore, we enunciate the results for an open coordinate set Ω in Rn, then we compose with diffeomorphism ψ−1 to see it in the manifold. Let Ω ⊂ Rnbe an open set and X be a smooth vector field on Ω. Fixed x in Ω, X induces a local one parameter group of transformations on Ω, {σX(t, x) = σ(t, x)}t which is the unique solution of the Cauchy problem
(10) ∂ σ(t, x) ∂t = X|σ(t,x) σ(0, x) = x.
This unique solution always exists for |t| sufficiently small. Moreover, if X = X(u1, · · · , ul) depends in smooth way on parameters (u1, · · · , ul) in an open set U ⊂ Rl and we consider compact sets L ⊂ U and K ⊂ Ω, there exists a constant 0 such that
1. DEFINITION OF SUB-RIEMANNIAN STRUCTURE 21
is a smooth function. Thanks to the uniqueness of (10), there holds σ(s, σ(t, x)) = σ(s + t, x) when x ∈ K, |s + t| < 0, (12)
σλX(t, x) = σX(λt, x) when x ∈ K, |λt| < 0. (13)
Now, by equation (12) the function x → σ(−t, x) is a C∞ inverse of x → σ(t, x). Therefore, x → σ(t, x) is a diffeomorphism on a compact set of Ω, for |t| sufficiently small. In this sense we construct a parameter group of diffeomorphisms.
Definition 1.10. We define the exponential mapping by exp(X)(x) = σX(1, x)
whenever the right hand side is defined.
For all t sufficiently small, σX(t, x) = σtX(1, x) = exp(tX)(x) is always well-defined. Now, let X1, · · · , Xl be smooth vector fields on Ω and (u1, · · · , ul) be parameters in Rk. Then, if (14) |u| = v u u t l X i=1 u2 i
is sufficiently small, |u| < 0, we have that the function
(15) (u1, · · · , ul, x) → exp l X i=1 ui Xi ! (x)
is well-defined and smooth. For further details see [29, Appendix]. Notice that if X a vector field in H on N
γ(t) = exp (tX) (p) = σX(t, p)
is a horizontal curve. Here, we denote in the same way the exponential mapping defined on Ω ⊂ Rn and its image through φ−1, where φ : U → Ω is a diffeomor-phism, U is an open set in N and p = φ(x).
Since the exponential mapping is a local diffeomorphism between the tangent space and the manifold, it makes sense the following definition
Definition 1.11. Given X1, · · · , Xk local frame ofH around p and commuta-tors Xk+1, · · · , Xnthat minimize the local homogeneous dimension Q. The canon-ical coordinates of q around p are the coefficients (u1, · · · , un) such that
q = exp n X i=1 ui Xi ! (p)
Let h be the metric making X1, · · · , Xk orthonormal, we can extend the norm (14), with l = k, to a homogeneous norm on the whole space
(16) kuk = n X i=1 |ui|Q/d(Xi) !1/Q . 2. Differential Operators
The notion of exponential mapping allow us to define the Lie derivative in the direction X.
Definition 1.12. Let X be a fixed vector field. We call Lie derivative of f in the direction of the vector X on the tangent space to N at a point p the derivative with respect to t of the function f (exp(tX)(p)) at t = 0.
Obviously, if f is C1 the Lie derivative d dt t=0f (exp(tX)(p)) = dfp d dt t=0exp(tX)(p) = dfp(Xp) = Xp(f )
is equal to the directional derivative Xf , but the Lie derivative can exist even if the directional derivative does not.
Definition 1.13. Let U ⊂ N be an open set. Let X1, · · · , Xk be a family of smooth vector fields defined on U and f : U → Rm. If the Lie derivatives Xjfi exist at p in U , for j = 1, · · · , k and i = 1, · · · , m, we define the horizontal Jacobian of f at p as the matrix:
JHf (p) = X1f1(p) · · · Xkf1(p) .. . . .. ... X1fm(p) · · · Xkfm(p) = ∇Hf1 .. . ∇Hfm .
A function f is of class CH1 if every element ofJHf is continuous with respect to the Carnot-Carathéodory distance (6). A function f is CH2 if every element of JHf is of class CH1. The space CHk is defined by induction.
Remark 1.3. Let f : U → Rm be a CE1 function. Then, JHf is the (1, 0) version of the restriction of the differential df toH, df|H, when the inner product h is the one which makes X1, . . . , Xk an orthonormal basis. Indeed, we have
df |H(Xj) = Xjf1 .. . Xjfm = h(grad(f1), Xj) .. . h(grad(fm), Xj) , where grad(fi) =Pk l=1ali Xl and we have Xjfi = h k X l=1 ali Xl, Xj ! = aji.
2. DIFFERENTIAL OPERATORS 23
Therefore, gradfi = ∇ Hfi.
Remark 1.4. Obviously, a function of class CH1 need not to be of class CE1. If X1, · · · , Xk satisfy the Hörmander rank condition with step s, then a function f of class Cs
H belongs to CE1. Hence, let X1, · · · , Xk be vector fields that satisfy the Hörmander rank condition, then
f ∈ CH∞ if and only if f ∈ CE∞.
Definition 1.14. Let U be an open set in N . A function f : U → Rl is differentiable at a point p ∈ U ⊂ N in the intrinsic sense if
fi(exp( n X j=1 ujXj)(p)) − fi(p) = k X j=1 ujXjfi(p) + o(kuk) i = 1, · · · , l, or in other words f (exp( n X j=1 ujXj))(p) − f (p) =JHf (p)(u1, · · · , uk)T + o(kuk)
Theorem 1.5. Let U ⊂ N be a connected open set and suppose that f : U → Rl is differentiable in the intrinsic sense at every point of p ∈ U . Fix p in U , let
q = exp( n X
j=1
ujXj)(p)
be a point next to p and ξ in Rl. Then there exists z in U such that hf (q) − f (p), ξi = hJHf (z)(u1, · · · , uk)T, ξi. Proof. F : [0, 1] → R, F (t) = hf (exp( n X j=1 tujXj)(x)), ξi.
F is continuous in [0, 1]. For every t ∈]0, 1[, F0(t) = d dt l X i=1 fi(exp( n X j=1 tejXj)(p)) ξi ! = l X i=1 lim t0→t 1 t0 fi(exp( n X j=1 t0ujXj)exp( n X j=1 tujXj)(p))+ − fi(exp( n X j=1 tujXj)(p)) ξi = l X i=1 k X j=1 ujXjfi(exp( n X j=1 tujXj)(p)) ξi = l X i=1 (JHf (exp( n X j=1 tejXj)(p))(u1, · · · , uk)t)i ξi.
By the mean value theorem, there exists τ ∈ [0, 1] such that F (1) − F (0) = F0(τ ). Hence, for z = exp
Pn
j=1τ ujXj
(p) there follows
hf (q) − f (p), ξi = F (1) − F (0) = hJHf (z)(u1, · · · , um)t, ξi.
Franchi, Serapioni and Serra Cassano gave a definition of a regular hypersur-faces in a Carnot group in [13, Definition 1.6]. A natural generalization of this definition is the following definition of a regular submanifold in a sub-Riemannian manifold.
Definition 1.15. A regular submanifold of dimension m is a subset of N that can be locally represented as the zero-set of a function f in C1
H(N, Rl), where l = n − m such the rank of JHf is equal to l.
However, in the present study we consider general smooth submanifolds in N and then we study the relation between their tangent space and the distribution by the notion of degree. Magnani, Vittone and Le Donne further expanded this approach in [24,26,27].
Here, we report an important result, the Rothschild-Stein’s Theorem of lifting and approximation [34, Theorem 5] that shows how it is possible to approximate general free up vector fields (i.e. the vector field of the distribution and their commutators up to step s are linear independents see [34]) with polynomial vector field generating a free Lie Group. The proof of this theorem is connected to
3. EXAMPLES AND APPLICATIONS 25
the Mitchell’s Theorem which shows that the tangent space to a sub-Riemannian manifold is its nilpotentization, which is a Carnot group.
Theorem 1.6 ( Rothschild-Stein’s theorem of lifting and approximation). Let H be a distribution generated by X1, · · · , Xk vector fields on N and let p be a point in N such that H
(i) verifies the Hörmander rank condition (ii) is free up to step s at p.
Choose Xk+1, · · · , Xn commutators such that X1, · · · , Xnspan the all tangent bun-dle, there they determine a canonical coordinates (u1, · · · , un) around p. Let G be the Carnot Group of step s with k generators and g its Lie algebra. Then there are Y1, · · · , Yn vector field in g and neighborhoods U of p in N and Ω of 0 in G with the following properties. We consider
Θ : U × U → Ω, Θ(ξ, η) = exp n X i=1 uiYi ! (0), where η = exp (Pn
i=1uiYi) (ξ). Therefore, if we fix ξ in U the mapping η → Θξ(η) = Θ(ξ, η) = (u1, · · · , un)
is a coordinate chart for U centered at ξ. In this chart Xi = Yi+ Ri i = 1, · · · , k where Ri is a differential operator of local degree 6 0.
3. Examples and applications
3.1. Heisenberg group. Let (H3, ∗) be a simply connected Lie group whose Lie algebra is
h= h1⊕ h2
where h1 = span{X, Y } and h2 = span{Z} with X, Y, Z satisfying the following bracket relations
(17) [X, Y ] = 2Z [X, Z] = 0 [Y, Z] = 0. We can identify h with R3, since the exponential mapping
exp : h −→ H3
is a global diffeomorphism. Indeed we can introduce global coordinates on H3 by
ϕ : R3 −→ H3
(x, y, t) 7→ exp(xX + yY + tZ)
Therefore, we can identify H3 with R3 and on R3 a pair of vector fields that satisfy (17) are
These vector fields lie on the kernel of the contact form w = dt − (xdy − ydx).
The group operation on R3 with these two vector fields and Z = ∂t is (x, y, t)· (x0, y0, t0) = (x + x0, y + y0, t + t0+ xy0− yx0).
Moreover, we have that (H3, h1, h) is a Carnot group, where h is a arbitrary metric on the distribution h1. Notice that the coefficient 2 in front of Z in (17) does not affect the structure of Carnot group and would change only the group operation providing a group isomorphic to the one just now presented . Therefore, we should define the Heisenberg group as the only one Carnot group of dimension three such that Lie algebra h = h1⊕ h2 satisfies the following conditions
rank(h1) = 2 and rank(h2) = 1.
Montgomery in [28] shows the connection between the Dido problem and the geodesics in Heisenberg group. We suggest the reader this lecture.
3.2. Rototraslation Group. Citti and Sarti in [9] proposed a model of low-level vision to mathematically model the functional structures of the primary visual cortex, they based their model on the previous works [20] by Hoffman and [32] by Petitot-Tondut where differential geometry models the visual cortex. To un-derstand this model we make a brief exposition of the functional architecture of the visual cortex, see for further details [22, Chapter 4]. The acquisition of the visual system starts in the retina, that after projects the information to the lateral geniculate nucleus and from there to the primary visual cortex V1. We can iden-tify the retinal structure with a plane R2. The primary visual cortex V1 processes the orientation via the simple cells and other features by complex cells (estima-tion of mo(estima-tion direc(estima-tion, detec(estima-tion of angles, curvature). The receptive field of a cell is the domain of the retinal plane to which the cell is connected with neural synapses of the retinal-geniculate-cortical path. When the receptive field of a cell is stimulated by a visual signal, the cell reacts generating spikes. On the recep-tive field there area “on”, if the spikes respond to posirecep-tive signal, and “off” area, if the spikes respond to negative signal. This behavior can be mathematically mod-eled by a function Ψ0 defined on the retinal plane. The retinotopic structure is a logarithmic conformal mapping between the retina and V1, that Citti and Sarti ignored in their study. Cortical cells are organized in columns corresponding to parameters as orientation, curvature, ocular dominance and color by the hyper-columnar structure. This structure, for simple cells, means that over each point of the retina there is a set of cells (hypercolumn) which are sensitive to all possible orientations. The non-maxima suppression selects the orientation of maximum output of the hypercolumn in response to a visual stimulus and suppresses all the others. There is also the connectivity structure that connects cells with the same orientation belonging to different hypercolumns.
3. EXAMPLES AND APPLICATIONS 27
Citti and Sarti consider a gray level image I on the retina as a real stimulus and they assume that over each point (x, y) in the retina plane the cells in the hypercolumn can code the direction of the level line of I. They assume that the cell in the hypercolumn, which gives a maximal response, is sensible to the direction of the level line of I in a point (x, y). The gradient ∇I = (Ix, Iy) is perpendicular to the level lines, then a tangent vector to a level line is (−Iy, Ix) or (Iy, −Ix). In order to save the information of direction they consider the angle θ(x, y) = −arctan(Ix, Iy), θ ∈ [0, π]. This process associates to each retinal point (x, y) a point (x, y, θ) in the three-dimensional space R2 × S1. Therefore, each level line γ(t) = (x(t), y(t)) is lifted to a curve ˜γ(t) = (x(t), y(t), θ(x(t), y(t))) in R2× S1. Notice that the vector field on the retinal plane tangent to the level lines of I, γ, at the point (x, y) is
(18) Xθ = cos(θ(x, y))∂x+ sin(θ(x, y))∂y.
A tangent vector on R2 × S1 to the lifted curve ˜γ is a linear combination of the vector field
X = cos(θ)∂x+ sin(θ)∂y, Y = ∂θ.
We define the distribution H = span{X, Y }, which is the kernel of the one-form ω = sin(θ)dx − cos(θ)dy.
The horizontal inner product h is the one which makes X and Y an orthonormal basis. Therefore, (S := R2×S1,H, h) is a contact sub-Riemannian manifold, which verifies the Hörmander rank condition. Indeed, we have
T = [X, Y ] = sin(θ)∂x− cos(θ)∂y and the rank of
cos(θ) sin(θ) 0 0 0 1 sin(θ) − cos(θ) 0
is three, since the matrix is invertible. One can think that each level line is lifted to space S separately, but the mechanism of non-maxima suppression is applied to the whole image producing a regular surface, see [9, 1.4.2] for the simple cell activity and [9, 1.6.1] in order to understand the non-maxima suppression. Therefore a image lifted to S is a regular surface in particular a θ-graph,
Gθ = {(x, y, θ) ∈ S : θ = θ(x, y)}.
In [9] they apply the iteration of the joint work of sub-Riemannian diffusion and non-maxima suppression to Gθin order to provide a succession of surfaces that con-verges to a minimal surface in the rototraslation space. This minimal surface would be the surface we elaborate in the visual cortex and allows to propagate existing information and to complete the boundaries. Thus, we understand the reason why it is useful to study the PDE for minimal surfaces in this sub-Riemannian setting, a topic deeply studied by Galli in [15] and Galli and Ritoré in [16].
If we consider a curve in the retinal plane γ(t) = (x(t), y(t)) with tangent vector Xθ defined in (18), it follows that γ is parametrized by the arc. Indeed,
| ˙γ(t)|2 = ˙x(t)2+ ˙y(t)2 = cos(θ)2+ sin(θ)2 = 1.
When we lift the curve γ to S, we obtain (˜γ)(t) = (x(t), y(t), θ(x(t), y(t)). The length of ˜γ is (19) l(˜γ) = Z b a p h(˜γ(t), ˜γ(t)) dt = Z b a q 1 + ˙θ(t)2 dt.
Notice that ˙θ(t) = k, where k is the curvature of the curve γ in the retinal plane. Let us remind that the elastica functional for a curve in the plane is
E(γ) =Z γ
√
1 + k2,
therefore, it follows that the length of a lifted curve ˜γ is equal to the elastica functional of γ on the plane.
Now, we show an application of the Rothschild-Stein’s Theorem at this simple case. We expand cos(θ) and sin(θ) at the first order at the point (x0, y0, θ0)
X = (cos(θ0) − sin(θ0)(θ − θ0)) ∂x+ (sin(θ0) + cos(θ0)(θ − θ0)) ∂y
| {z } ˜ X + o(|θ − θ0|) ∂x+ o(|θ − θ0|) ∂y | {z } R , Y =∂θ. Now, we have ˜
X = (cos(θ0) − sin(θ0)(θ − θ0)) ∂x+ (sin(θ0) + cos(θ0)(θ − θ0)) ∂y, Y =∂θ,
[ ˜X, Y ] = − sin(θ0)∂x+ cos(θ0)∂y.
Let us consider the following transformation, that is essentially the exponential mapping in S, (20) x0 y0 θ0 = cos(θ0) x + sin(θ0)y − sin(θ0) x + cos(θ0)y θ − θ0 and ∂ ∂x = cos(θ0) ∂ ∂x0 − sin(θ0) ∂ ∂y0 ∂ ∂y = sin(θ0) ∂ ∂x0 + cos(θ0) ∂ ∂y0.
3. EXAMPLES AND APPLICATIONS 29
In these new coordinates
˜
X =∂x0+ θ0 ∂y0,
Y =∂θ0,
[ ˜X, Y ] = − ∂y0.
This is the Heisenberg algebra.
3.3. Engel group. The Engel group E is a simply connected Carnot group whose Lie algebra is
e = V1⊕ V2⊕ V3 where
rank(V1) = 2, rank(V2) = 1 and rank(V3) = 1. Since the exponential is a global diffeomorphism
exp : e −→ E we can represent the Engel group by R4 where
X1 = ∂x1 − x3∂x2 − x4∂x3 and X2 = ∂x4 that generate V1,
X3 = [X1, X2] = ∂x3 that generates V2,
X4 = [X1, [X1, X2]] = ∂x2 that generates V3.
This representation will be useful in the following section to show that this group is the tangent space to a four dimensional Engel sub-Riemannian manifold. Whereas Le Donne and Magnani present a more standard representation in [24] where
X1 = ∂x1, X2 = ∂x2 + x1∂x3 + x2 1 2 ∂x4, X3 = [X1, X2] = ∂x3 + x1∂x4, X4 = [X1, [X1, X2]] = ∂x4.
3.4. Curvature and orientation. Let E = R2× S1× R and let (21) X1 = cos(θ)∂x+ sin(θ)∂y+ k∂θ, X2 = ∂k
be vector fields on E, we set H = span{X1, X2}. To define a sub-Riemannian manifolds we need an inner product on the distribution H. In the present work we will use two different metrics on the distribution H: h1, the one which makes X1 and X2 orthonormal, and h2, the one induced by the Euclidean metric
(22) h1 = 1 0 0 1 h2 = 1 + k2 0 0 1 .
Therefore, (E,H, h1) and (E,H, h2) are sub-Riemannian manifolds, we will specify the metric we use. These vector fields satisfy the Hörmander rank condition. Indeed, we have
X3 = [X1, X2] = −∂θ (23)
X4 = [X1, [X1, X2]] = − sin(θ)∂x+ cos(θ)∂y. (24)
We are interested in studying minimal surface in this setting, in particular (θ,κ)-graphs. Therefore, we need the implicit function theorem to provide a man-ageable parametrization. It would be a problem if we considered the definition of regular surface adopted in [13]. Indeed we should take into account
JHf =
X1f1 X2f1 X1f2 X2f2
.
Magnani in [26] deal with implicit function theorem in stratified groups with re-spect to the intrinsic notion differentiability. However, we consider smooth sub-manifolds. Therefore our implicit function theorem is standard.
Theorem 1.7 (Implicit Function Theorem). Let Ω ⊂ E be an open set and let f : Ω → R2 be a continuous and CH2(Ω, R2) function. If
Σ = {ξ = (x, y, θ, k) ∈ Ω : f (x, y, θ, k) = 0} and suppose that
(25) det X2f 1 X 3f1 X2f2 X3f2 ( ¯ξ) 6= 0. Then there exist neighborhoods I, J ⊂ R2 such that
Σ ∩ (I × J ) = {(x, y, u1(x, y), u2(x, y)) : (x, y) ∈ I}.
Proof. First of all, notice that f in CH2(Ω, R2) implies CE1(Ω, R2) by Remark
1.4. Let (R3×]0, 2π[, 1 × eiθ = ψ
1) and (R3×] − π, π[, 1 × ei(π+θ) = ψ2) be the two cards of E. If we want to express a point p of E we can use the coordinates
ψi−1 : Ui → Wi, ψi−1(p) = (x, y, θ, k) i = 1, 2 where we have set
U1 = R3× S/(1,0)1 , W1 = R3×]0, 2π[, U2 = R3× S/(−1,0)1 W2 = R3×] − π, π[. Now we have X2 = ∂k, X3 = −∂θ and we can define the transformation
G(x, y, θ, k) = (x, y, −θ, k). With this choice it follows
3. EXAMPLES AND APPLICATIONS 31
Therefore the condition (25) is equivalent to (26) det ∂θ(f ◦ G) 1 ∂ k(f ◦ G)1 ∂θ(f ◦ G)2 ∂k(f ◦ G)2 (¯x, ¯y, −¯θ, ¯k) 6= 0.
Now thanks to the classic implicit function theorem, there exist neighborhoods I of (¯x, ¯y) and ˜u1 : I → R and u2 : I → R such that
(f ◦ G)(x, y, ˜u1(x, y), u2(x, y)) = 0. Therefore we set u1 = − ˜u1 it follows
f (x, y, u1(x, y), u2(x, y)) = 0
and the proof is complete.
Now, we show that the Carnot group that approximates the structure of E is the Engel group. As we did in (3.2), we want to deduce approximate vector fields for this structure. Then, we expand at the first order sin(θ) and cos(θ) around θ0
X1 = (cos(θ0) − sin(θ0)(θ − θ0)) ∂x+ (sin(θ0) + cos(θ0)(θ − θ0)) ∂y+ k ∂θ
| {z } Y1 + o(|θ − θ0|) ∂x+ o(|θ − θ0|) ∂y | {z } R , X2 =∂k. Therefore, we have [X1, X2] = −∂θ [X1, [X1, X2]] = sin(θ0)∂x− cos(θ0)∂y | {z } Y4 +o(1)∂x+ o(1)∂y.
If we cut at the first order, we obtain the following structure
Y1 = cos(θ0) − sin(θ0)(θ − θ0) ∂x+ sin(θ0) + cos(θ0)(θ − θ0) ∂y + k ∂θ, Y2 = ∂k,
Y3 = −∂θ,
Y4 = − sin(θ0)∂x+ cos(θ0)∂y. Under the following transformations
(27) x0 y0 θ0 k0 = cos(θ0) x + sin(θ0)y − sin(θ0) x + cos(θ0)y θ − θ0 k .
Renaming (x0, y0, θ0, k0) = (x, y, θ, k), we obtain
Y1 = ∂x+ θ∂y+ k∂θ, Y2 = ∂k, Y3 = −∂θ,
Y4 = ∂y.
This algebra generates a four dimensional Carnot group which is known as the Engel group.
In the sub-Riemannian manifold E we have already presented we have X1 = X + k Y,
where X and Y are the vector field of S. We can lift a horizontal curve γ = (x(t), y(t), θ(t)) in S to a curve ¯γ = (x(t), y(t), θ(t), ˙θ(t)) in E. Therefore we have
˙¯ γ(t) = ( ˙x(t), ˙y(t), ˙θ(t), ¨θ(t)) = X1+ ¨θ(t)X2, | ˙¯γ(t)|2h1 = 1 + ¨θ(t)2, | ˙¯γ(t)|2h 2 = 1 + ˙θ(t) 2 + ¨θ(t)2, and the length of the curve ¯γ is
lhi(¯γ) =
Z b
a
| ˙¯γ(t)|hi dt.
After all, to know the curvature Ks(γ) of the horizontal curve γ in S we have to derive the orthonormal tangent vector
E1(t) = X(t) + ˙θ(t)Y q 1 + ˙θ(t)2 . Hence, it follows dE1 dt (t) = ¨ θ(t) 1 + ˙θ(t)2 Y − ˙θ(t)X q 1 + ˙θ(t)2 + ˙ θ(t) q 1 + ˙θ(t)2 T.
We have only the metric h1 on the distribution, therefore for us the orthonormal to E1 is
N = qY − ˙θ(t)X 1 + ˙θ(t)2 and the curvature for the horizontal curve is
(28) KS(γ) =
¨ θ(t) 1 + ˙θ(t)2.
3. EXAMPLES AND APPLICATIONS 33
In this setting the elastica functional on a horizontal curve in S is not equal to the length of the lifted curve in E. Therefore, in order to deduce the previous property we substitute the vector field X1 and X2 by
Z1 = cos(ϕ)X + sin(ϕ)Y Z2 = ∂ϕ
where we have set k = tan(ϕ). In this way, if we set H = span{Z1, Z2} and we choose as horizontal metric h3 the one making Z1 and Z2 orthonormal, we have a sub-Riemannian manifold
( ˜E = R2× S1× S1,H, h3)
where H is bracket-generating distribution. Indeed, there holds Z3 = [Z1, Z2] = sin(ϕ)X − cos(ϕ)Y
Z4 = [Z1, Z3] = − sin(θ)∂x+ cos(θ)∂y and Z1, · · · , Z4 are linear independents
det
cos(ϕ) cos(θ) cos(ϕ) sin(θ) sin(ϕ) 0
0 0 0 1
sin(ϕ) cos(θ) sin(ϕ) sin(θ) − cos(ϕ) 0
− sin(θ) cos(θ) 0 0 6= 0.
Here, if we consider a lifted curve γ form R2 to S ϕ is the real number in ] − π, π[ such that
˙γ(t) = cos(ϕ)X + sin(ϕ)Y.
Therefore, the tangent vector to a lifted curve ¯γ(t) = (γ(t), ϕ(t)) is | ˙¯γ(t)|2 = 1 + ˙ϕ(t)2
and its length
l(¯γ(t)) = Z b
a p
1 + ˙ϕ(t)2. Now, we set ϕ(t) = arctan( ˙θ(t)), thus
˙ ϕ(t) =
¨ θ(t) 1 + ˙θ(t)2,
that is equal to (28). Hence, the length of a lifted curve ¯γ = (γ, ϕ) in ˜E is equal to the elastica functional in S evaluate in γ
E(γ) =Z b a 1 + ¨ θ(t) 1 + ˙θ(t)2 !2 1 2 dt.
Remark 1.5. Notice that locally k = tan(ϕ) is a change of coordinates. There-fore we can see ˜E locally as E with an other metric h3. Indeed, under this change of coordinates
X1 = 1
cos(ϕ)(cos(ϕ)X + sin(ϕ)Y ), X2 =
1
cos(ϕ)2 ∂ϕ.
Therefore, h3 on the distribution generated by X1, X2 should be h3 =
cos(ϕ)2 0 0 cos(ϕ)4
.
In these coordinates the commutators change but they are proportional to Z3 and Z4 where the factors depend only on ϕ.
CHAPTER 2
Area in a sub-Riemannian manifold
Our purpose in this chapter is to give a general definition of the area in a sub-Riemannian manifold for a submanifold of arbitrary dimension using the notion of degree studied by Magnani-Vittone in [27] and Le Donne-Magnani in [24]. In this chapter we prove basic properties of the degree and the area, then we show that our definition of area for a hypersurface in the Heisenberg group coincides with the one used in [33] and we provide a suitable definition of area for a surface in S and E. Finally, in the last section we compare the Hausdorff dimension of a submanifold to the notion of degree.
Let N be a smooth manifold and n be the dimension of N . Let H be a distribution on N and U be an open subset of N . Locally {X1, · · · , Xk} span H on the open set U . The distribution H is a subbundle of constant dimension k of the tangent space T U , see [28]. Moreover, h is a metric defined only on the subbundle H. Therefore, (N, H, h) has a structure of sub-Riemannian manifold and furthermore we suppose that X1, · · · , Xkverify the Hörmander rank condition. The Lie brackets of vector fields in H generate a flag of subbundles
(29) H ⊂ H2 ⊂ · · · ⊂Hr⊂ · · · ⊂ T N, with H2 =H + [H, H], Hr+1 =Hr+ [H, Hr], where [H, Hk] = {[X, Y ] : X ∈H, Y ∈ Hk}.
The fact that X1, · · · , Xk verify Hörmander rank condition is equivalent, at least in the case of N compact, to the assumption that there is an s such thatHs = T N . Henceforth, we will do this assumption. Here, we follow Montgomery’s book [28, 2.3]. The flag of subbundles at a point p is a flag of subspaces of TpN
(30) Hp ⊂H2p ⊂ · · · ⊂H s
p = TpN
and we set ni(p) = dimHip. The integer list (n1(p), · · · , ns(p)) of dimensions is called the growth vector of H at p. Moreover, the smallest s such that Hs
p = TpN is called the step of the distribution H at the point p.
Definition 2.1. A distribution H on a manifold N is regular at a point p in N if the growth vector is constant in a neighborhood of p.
Example 2.1. In order to show an example of sub-Riemannian manifold with not regular points, we consider the Grˇusin plane G2. Let us consider the plane R2 with coordinates x, y and the sub-Riemannian metric which makes orthonormal the vector fields
X1 = 1 0 , X2 = 0 x .
These vector fields span all the tangent space, except along the line x = 0. There, if we add the Lie bracket
[X1, X2] = 0
1
,
the distribution verifies the Hörmander rank condition and thus the hypothesis of the Chow Theorem. Outside the line x = 0, the sub-Riemannian metric is
ds = dx2 + 1 x2dy
2,
that it is essentially a Riemannian metric. Now, if we consider a point p in {(x, y) ∈ R2 : x 6= 0} then we can find a sufficiently small neighborhood where the dimension of the growth vector n1(p) is constantly equal to two. Hence, p is a regular point. If we suppose that p ∈ {(x, y) ∈ R2 : x = 0} the growth vector will be (n1(p), n2(p)) = (1, 1). Then an open neighborhood of p has to intersect the set where x 6= 0, and so the growth vector is equal to n1(p) = 2. Therefore, for each neighborhood of a point in the line x = 0, we have that the growth is not con-stant. Hence, this is not a regular point. We suggest the reader to see [2, page 31] for an example of singular point where the dimension of the first layer is constant. There the sub-Riemaniann structure is R3 equipped of the distribution
H = span{X1 = ∂x+ 1 2y
2∂
z, X2 = ∂y}.
Each point in the surface π = {y = 0} is a singular point. Indeed, the growth vector for p in π is n(p) = (2, 2, 3) and for p not in π is n(p) = (2, 3), since
[X1, X2] = −y∂z [X2, [X1, X2]] = −∂z. We set Hi =Hi/Hi−1 and define
(31) Gr(H) = H ⊕ H2/H ⊕ · · · ⊕ Hs/Hs−1 = H1⊕ · · · ⊕ Hs,
which is the graded bundle corresponding to the flag of bundlesHi. When p is a regular point for the distribution, we call Gr(H))(p) the nilpotentization of H at p.
Definition 2.2. Given a set of vector fields X1, · · · , Xk ∈ T N and a multi-index
J = (j1, · · · , jl) ∈ {1, · · · , k}l, we set
1. DEGREE OF A SUBMANIFOLD IN A SUB-RIEMANNIAN MANIFOLD 37
We say that XJ is the commutator of length l of X1, · · · , Xk if XJ ∈ Hl and XJ ∈/Hl−1.
Let U be a neighborhood of a regular point p in N , the tangent bundle T U can be written as T U = H1⊕ · · · ⊕ Hs, where H1 = span{X1, · · · , Xk}, H2 = span{[Xi, Xj] i 6= j i, j = 1, · · · , k : [Xi, Xj] /∈ H1} = span{Xk+1, · · · , Xn2}, .. .
Hl= span{ XJ a commutator of length l : J ∈ {1, · · · , k}l} = span{Xnl−1+1, · · · , Xnl},
.. .
Hs= span{ XJ a commutator of length s : J ∈ {1, · · · , k}s} = span{Xns−1+1, · · · , Xn}.
Where s is the step defined in 1.6. A frame
(X1, · · · , Xk, Xn1+1· · · , Xn2, · · · , Xns−1+1, · · · , Xn)
is an adapted basis to the flag (30) at each point.
Definition 2.3. An order basis (v1, · · · , vn) is said to be adapted to a flag V1 ⊂ V2 ⊂ · · · ⊂ Vs
if the first di = dim(Vi) vectors form a basis Vi.
For each Xi in the adapted basis we can assign its length, which is also called weight. The assignment i 7→ wi is called the weighting associated to the growth vector.
Moreover, we assume that the dimension dim Hi(p) is constant in p in N for each i = 1, · · · , s, which is known as the equiregularity assumption, for further details see [19, page 95].
1. Degree of a submanifold in a sub-Riemannian manifold
Definition 2.4. Let (X1, X2, · · · , Xn) be an adapted basis to the flag of T N . The degree d(j) of Xj is the unique integer r such that Xj ∈ Hr. Let
be a simple m-vector of Λm(N ), where J = (j1, j2, · · · , jm) and 1 6 j1 < j2 < · · · < jm 6 n. The degree of XJ is the integer d(J ) defined by the sum d(j1)+· · ·+d(jm).
Let
τ =X J
τJXJ
be a m-vector in Λm(N ) represented with respect to the fixed adapted basis (X1, · · · , Xn) where τJ are functions. The degree of τ is defined as the integer
d(τ ) = max{d(XJ) ∈ N : τJ 6= 0}.
Let M be a manifold such that dim(M ) = m < n = dim(N ) and let Φ : M → N be an embedding, which is defined to be an injective immersion which is an homeomorphism onto its image. We have that dΦ(TpM ) ⊂ TΦ(p)N for each p in M and that the dimension dim(dΦ(TpM )) = m, due to the differential dΦ being injective. We set Σ = Φ(M ). Therefore, the degree of a point p in M is defined as
dΣ(p) = d(τΣ(p)), where
τΣ(p) ∈ {v1∧ · · · ∧ vm : B = (v1, · · · , vm) a basis of dΦ(TpM )}.
Obviously, the degree is independent of the choice of the basis of the tangent subspace. Indeed, if we consider another basis B0 = (v10, · · · , v0m) of dΦ(TpM ), there holds
v1∧ · · · ∧ vm = det(MB,B0) v01∧ · · · ∧ vm0 .
Since det(MB,B0) 6= 0, the degree is well-defined.
Definition 2.5. The projection of τ onto m-vectors of degree r is defined by (τ )r =
X
d(J )=r τJXJ.
The degree d(Σ) of a submanifold Σ is the integer max
p∈Σ dΣ(p).
2. Equivalence between our degree and Gromov’s degree
Mikhael Gromov gave a definition of degree in [19, 0.6.B], we want to show that his definition is equivalent to ours given in Section 1of this Chapter.
Definition 2.6 (Gromov’s degree). Let M be a submanifold of an equiregular sub-Riemannian manifold N equipped with its flag of subbundles
2. EQUIVALENCE BETWEEN OUR DEGREE AND GROMOV’S DEGREE 39 We set ˜Hj p = TpM ∩Hpj, then it follows ˜ H1 p ⊂ ˜H 2 p ⊂ · · · ⊂ ˜H s p = TpM. We denote ˜mj = dim( ˜Hpj/ ˜Hj−1p ), then we set
˜ DH(p) = s X j=1 j ˜mj.
This definition of degree is much more geometrical than the one we gave before, because it expresses the intersection between the tangent space of the submanifold and the flag of subbudles. The intersection of the tangent space TpM and the flag of subbundles will be constant for k1 subbudles and it will change at k1+ 1, then it will be constant until the subbudle k2 and then it will change at k2 + 1 and then again constant until subbudle k3. Iterating this process, we obtain the finite sequence k1, · · · , kr ∈ {1, · · · , s − 1}. Therefore we have
M1 = TpM ∩H1 = · · · = TpM ∩Hk1 ( M2 = TpM ∩Hk1+1 = · · · = TpM ∩Hk2 (
.. .
Mr = TpM ∩Hkr+1 = · · · = TpM ∩Hs,
where we set Li = dim(Mi) i = 1, · · · , r. Obviously,
M1 ⊂ M2 ⊂ · · · ⊂ Mr = TpM, L1 < L2 < · · · < Lr= m.
Now, we can choose v1, · · · , vL1, vL1+1, · · · , vL2, · · · , vLr−1+1, · · · , vLr a basis of the
tangent space TpM such that
v1, · · · , vL1 ∈ M1 vL1+1, · · · , vL2 ∈ M2r M1 .. . vLi−1+1, · · · , vLi ∈ Mir Mi−1 .. . vLr−1+1, · · · , vLr ∈ Mrr Mr−1.
If v is a vector in {vLi−1+1, · · · , vLi} then v belongs to H
ki−1+1
r Hki−1. Therefore,
it follows that the degree of the m-vector is equal to
d(v1∧ · · · ∧ vL1 ∧ vL1+1∧ · · · ∧ vL2 ∧ · · · ∧ vLr−1+1∧ · · · ∧ vLr) =
L1+ (L2− L1)(k1+ 1) + · · · + (Li− Li−1)(ki−1+ 1)+ + · · · + (Lr− Lr−1)(kr−1+ 1).
Now, let us compute Gromov’s degree. In order to determine it we recall that ˜
mj = dim( ˜Hjp/ ˜Hj−1p ), where ˜Hjp = TpM ∩Hjp. Thus, we have
˜ m1 = L1, ˜m2 = 0, · · · , ˜mk1 = 0, ˜ mk1+1 = L2− L1, ˜mk1+2= 0, · · · , ˜mk2 = 0, .. . ˜
mki−1+1 = Li− Li−1, ˜mki−1+2 = 0, · · · , ˜mki = 0,
.. . ˜
mkr−1+1 = Lr− Lr−1, ˜mkr−1+2 = 0, · · · , ˜mkr = 0.
Therefore, the Gromov’s degree at the point p is ˜
DH(p) =L1+ (L2− L1)(k1+ 1) + · · · + (Li− Li−1)(ki−1+ 1) + · · · + (Lr− Lr−1)(kr−1+ 1).
Hence, the two definitions of degree are equivalent.
Remark 2.1. In [25] Magnani writes about horizontal and non-horizontal points and also about the degree of a submanifold in a Carnot group. He sets that p in Σ is a non-horizontal point when TpΣ and Hp are transversal and p is horizontal when these subspaces are not transversal. In other words, a point p is horizontal if
Hp− dim(TpΣ ∩Hp) < k.
Overall, the notion of degree is more sophisticated than the one of horizontal point, because the degree detects all possible intersections between the tangent space and each layer of the distribution, not only between the tangent space and the distribution.
3. Semicontinuity of the degree
Here our aim is to prove that the degree of a vector field on a sub-Riemaniann manifold (N,H), defined in2.4, is lower semicontinuous at a regular point p in N . Let U ⊂ N be an open neighborhood of p and let
(32) v(q) = s X i=1 ni X j=ni−1 cij(q)(Xj)q
4. SUB-RIEMANIANN AREA OF A SUBMANIFOLD 41
be a smooth vector field on U , where (X1, · · · , Xn) is an adapted basis to the flag of the tangent space generated by the bracket-generating distribution
H = span{X1, · · · , Xk}.
In (32) we adopt the convention that n0 = 1. Let d be an integer number such that cdk(p) 6= 0 where k is a integer in {nd−1, · · · , nd} and cij(p) = 0 when i = d + 1, · · · , s. Therefore, the degree d(v(p)) of v at p is equal to d. Since coefficients are continuous, there exists U0 ⊂ U neighborhood such that cdk(q) 6= 0 for each q in U0. Therefore for each q in U0 the degree of v(q) is greater than or equal to the degree of v(p),
d(v(q)) > d(v(p)) = d.
The degree at q could be strictly greater than d if there is a coefficient cij(p) with i = d + 1, · · · , s that is equal to zero at p but over U0 is different to zero. Hence, we have
lim inf
q→p d(v(q)) > d(v(p)).
4. Sub-Riemaniann area of a submanifold
We extend the metric h defined on H to a Riemaniann metric g such that g|H = h and the spaces Hi(p) are orthogonal for each p in N . Now, let r > 0 be a real number and we will consider the Riemannian metrics gr such that
(33) gr(Xi, Xj) =
rd(Xi)+d(Xj )−22
−1
g(Xi, Xj) i, j = 1, · · · , n,
where Xi and Xj belongs to the adapted basis (X1, · · · , Xn). We will consider the m-vector fields ˜ XJ = rd(Xj1 )−1 2 Xj 1 ∧ · · · ∧ rd(Xjm )−12 Xj m , J = (j1, j2, · · · , jm), 1 6 j1 < · · · < jm 6 n. Let (U, ϕ = y1, · · · , ym) be local coordinates in p ∈ M and ∂
∂y1 p , · · · , ∂ ∂ym p ! is a basis of TpM . Furthermore, B = dΦ ∂ ∂y1 p ! , · · · , dΦ ∂ ∂ym p !!
is a basis of dΦ(TpM ) and we can express dΦ ∂ ∂y1 p ! ∧ · · · ∧ dΦ ∂ ∂ym p ! =X J ˜ τJ X˜J
with respect to the basis { ˜XJ : J = (j1, · · · , jm)}. We can take into account the Jacobian matrix (gij) which is equal to
gr dΦ ∂ ∂y1 p ! , dΦ ∂ ∂y1 p !! · · · gr dΦ ∂ ∂y1 p ! , dΦ ∂ ∂ym p !! .. . . .. ... gr dΦ ∂ ∂ym p ! , dΦ ∂ ∂y1 p !! · · · gr dΦ ∂ ∂ym p ! , dΦ ∂ ∂ym p !! .
Now, we will define a metric on the m-vectors using the metric gr. Let e1, · · · , em and e01, · · · , e0m be vectors in dΦ(TpM ). Thanks to the metric we can define the one-forms wi(v) = gr(v, ei) ∀ v ∈ dΦ(TpM ), i = 1, · · · , m. We set (34) gr(e1∧ · · · ∧ em, e01∧ · · · ∧ e 0 m) = (w1∧ · · · ∧ wm)(e01, · · · , e 0 m) = X σ∈Sm sgn(σ)(w1⊗ · · · ⊗ wm)(e0σ(1), · · · , e 0 σ(m)) = X σ∈Sm sgn(σ)gr(e0σ(1), e1) · · · gr(e0σ(m), em). Therefore, by (34) and the Leibniz formula for the determinant there follows
dΦ ∂ ∂y1 p ! ∧ · · · ∧ dΦ ∂ ∂ym p ! 2 gr = det(gij)(p).
Let {(Uα, ϕα)}α∈I be an atlas of the manifold M and let {Ψα}α∈I be a partition of unity subordinated to the cover {Uα} such that the compact supports of Ψα are completely contained in Uα. Therefore the Riemannian area is
area(Φ(M ), gr) = X α∈I Z ϕ(Uα) d(Φ ◦ Ψα) ∂ ∂y1 ϕ−1α (ξ) ! ∧ · · · · · · ∧ d(Φ ◦ Ψα) ∂ ∂ym ϕ−1α (ξ) ! gr ◦ ϕ−1α (ξ) dξ,
where ξ ∈ Rm and dξ = dξ1· · · dξm. Notice that here we consider the Lebesgue measure on the chart, but the same argument holds if M is equipped with a different measure µ. When we consider the measure µ we will compute the integral respect dµ instead of dξ.
If we set
5. AN INTERESTING CASE OF SUB-RIEMANIAN AREA 43
the submanifold Φ(M ) can be written as
Φ(M ) = Φ(M )m∪ · · · ∪ Φ(M )d,
where d = d(Φ(M )) is the degree of the submanifold. Finally, we define the sub-Riemannian area of Φ(M ) as (35) A(Φ(M ), g) = lim r→0 d X i=m ri−m2 area(Φ(M )i, gr).
Remark 2.2. Let (M, µ) a manifold of degree d embedded in N . Notice that only coefficients of m-vector fields of degree d survive because when r tends to zero the metric factor depending on the degree neutralize the factor rd−k2 that multiplies
the Riemannian area. Therefore, we have A(Φ(M ), g) =X α∈I Z ϕ(Uα) d(Φ ◦ Ψα) ∂ ∂y1 ϕ−1α (ξ) ! ∧ · · · · · · ∧ d(Φ ◦ Ψα) ∂ ∂ym ϕ−1α (ξ) ! ! d g ◦ ϕ−1α (ξ) dµ,
where | · |g denotes the norm on the m-vector induced by g and (·)d denotes the projection on the m-vector of degree d.
5. An interesting case of sub-Riemanian area
In general the sub-Riemanian area is dependent of the metric extension of the horizontal metric h.
Example 2.2. Let H1 ⊗ H1 be the direct product of Heisenberg space where we consider real coordinates (x, y, z, x0, y0, z0) and the Lie algebra is generated by
X = ∂x− y∂z, Y = ∂y+ x∂z, Z = ∂z, X0 = ∂x0− y0∂z0, Y0 = ∂y0 + x∂z0, Z0 = ∂0
z, and the only commutator relations not null are
[X, Y ] = 2Z [X0, Y0] = 2Z0.
Therefore we have 4-dimensional distribution H generated by X, Y, X0, Y0 and let h the horizontal metric making X, Y, X0, Y0 an orthonormal basis. Let Ω be a bounded open set of R2, there we consider the surface Σ parametrized by
Φ : Ω −→ H1⊗ H1
(s, t) → (s, 0, u(s, t), 0, t, u(s, t))
where u is a smooth function such that ut(s, t) ≡ 0. Therefore, it follows Φs =(1, 0, us, 0, 0, us) = X + us Z + us Z0,
and we have
Φs∧ Φt= X ∧ Y0+ us(Z ∧ Y0 + Z0 ∧ Y0).
When us(s, t) is different from zero the degree is three. Now, we consider the following metrics gλ,ν = I4 0 0 0 λ 0 0 0 ν
on the basis (X, Y, X0, Y0, Z, Z0). These metrics extend the horizontal metric and make H1 and H2 spaces orthogonal. The sub-Riemannian area of Σ depends on the metric elected, indeed
A(Σ, gλ,ν) = Z
Ω
us(λ + ν) dxdy.
Therefore, the PDE for the minimal surfaces obtain by the first variation of the area functional depends on the metric extension of h.
However, let d = d(Σ) be the degree of the submanifold of dimension m. We assume that m-vector
v := dΦ ∂ ∂y1 p ! ∧ · · · ∧ dΦ ∂ ∂ym p !
is expressed as the m-vector of an adapted basis, we suppose that the terms of degree d are a wedge product of m − 1 vectors of the first layer H1 and one vector of Hd−m+1. Moreover, we suppose that the Hd−m+1 has dimension one. We will call all these assumptions the HC hypothesis. In the definition of this condition we have been inspired by the fact that in the Heisenberg group, in the visual cortex S and E the required assumptions are satisfied.
Here, we assume the HC hypothesis. Therefore, let Xi1, · · · , Xim−1 be elements
of the basis in the first layer and Xm be the only vector of the basis in Hd−m+1 and the terms of degree d of v are
l X
i=1
aiXi1 ∧ · · · ∧ Xim−1∧ Xm
If we apply the metric gr extended to the m-vectors (36)
gr(Xi1 ∧ · · · ∧ Xim−1∧ Xim, Xi1 ∧ · · · ∧ Xim−1∧ Xim)
= X
σ∈Sm
sgn(σ)gr(Xσ(i1), Xi1) · · · gr(Xσ(im−1), Xim−1)gr(Xσ(m), Xm)
Since the metric makes the layer orthogonal the permutation σ must fix the last index m, i.e. σ(m) = m. Let g and ¯g be two metrics such that g|H = h, ¯g|H = h, then there exists λ a positive real number such that
gr(Xm, Xm) = λ¯gr(Xm, Xm) = λ
